Cultural change constitutes a Darwinian evolutionary process, comprising the three Darwinian principles of variation, selection and inheritance. Yet cultural evolution is not identical to genetic evolution: the sources of variation, the forms of selection and the modes of inheritance found in cultural evolution may be very different to those found in genetic evolution. Here, I review research conducted in the last 30 years that has built a Darwinian theory of cultural change by borrowing the rigorous, quantitative methods developed by biologists to explain biological evolution, yet simultaneously acknowledging the differences between cultural and genetic evolution. I argue that the quantitative nature of Darwinian methods (e.g. statistical analysis, formal models, laboratory experiments) has resulted in a significantly better understanding of cultural phenomena than many traditional non-evolutionary, non-scientific approaches to cultural change in the social sciences and humanities. Evolutionary theory also provides a synthetic framework within which different branches of the social sciences and humanities may be integrated, equivalent to the ‘evolutionary synthesis’ that integrated the biological sciences in the early 20th century.